Spectral noncommutative geometry and quantization: a simple example

Details Spectral noncommutative geometry and quantization: a simple example Spectral noncommutative geometry and quantization: a simple example

1999-04-13

arxiv.org/abs/gr-qc/9904029v1

Phys.Rev.Lett. 83 (1999) 1079-1083

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

7 pages, no figures

Scientific paper

10.1103/PhysRevLett.83.1079

We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over this phase space and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two ``spacetime points''), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operator *D on the direct product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization of the family of the triples (A, H, D).

Affiliated with

Also associated with

No associations

Rating

Spectral noncommutative geometry and quantization: a simple example does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Spectral noncommutative geometry and quantization: a simple example, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Spectral noncommutative geometry and quantization: a simple example will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-122808